Global existence for energy critical waves in 3-d domains

Burq, Nicolas ¹   ; Lebeau, Gilles ²   ; Planchon, Fabrice ³

¹ Laboratoire de Mathématiques, Université Paris Sud, UMR 8628 du C.N.R.S., Bât 425, 91405 Orsay Cedex, France and Institut Universitaire de France
² Laboratoire J.-A. Dieudonné, UMR 6621 du C.N.R.S, Université de Nice - Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France and Institut Universitaire de France
³ Laboratoire Analyse, Géométrie & Applications, UMR 7539 du C.N.R.S, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France

Journal of the American Mathematical Society, Tome 21 (2008) no. 3, pp. 831-845

Résumé

We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(\Omega ) \times L^2( \Omega )$ for any smooth (compact) domain $\Omega \subset \mathbb {R}^3$. The main ingredient in the proof is an $L^5$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.

Détail

DOI : 10.1090/S0894-0347-08-00596-1

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